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In this paper, we obtain a bulk dual to SYK model, including SYK model with$U(1)$ charge, by Kaluza-Klein (KK) reduction from three dimensions. We showthat KK reduction of the 3D Einstein action plus its boundary term gives theJackiw-Teitelboim (JT) model in 2D with the appropriate 1D boundary term. Thesize of the KK radius gets identified with the value of the dilaton in theresulting near-AdS$_2$ geometry. In presence of U(1) charge, the 3D modeladditionally includes a $U(1)$ Chern-Simons (CS) action. In order to describe aboundary theory with non-zero chemical potential, we also introduce a couplingbetween CS gauge field and bulk gravity. The 3D CS action plus the new couplingterm with appropriate boundary terms reduce in two dimensions to a BF-typeaction plus a source term and boundary terms. The KK reduced 2D theoryrepresents the soft sector of the charged SYK model. The pseudo-Nambu-Goldstonemodes of combined $\textit{Diff} /\mathbb{SL}(2,\mathbb{R})$ and $U(1)_{\rmlocal}/U(1)$ transformations are represented by combined large diffeomorphismsand large gauge transformations. The effective action of the former isreproduced by the action cost of the latter in the bulk dual, after appropriateidentification of parameters. We compute chaotic correlators from the bulk andreproduce the result that the contribution from the "boundary photons"corresponds to zero Liapunov exponent.

Holographic dual to charged SYK from 3D Gravity and Chern-Simons